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 Alligation or Mixture :-
 ::Important Formulas ::Examples

Direct Formula / Rule 11 :

• Theorem: A trader has N kg of a certain item, a part of which he sells at x% profit and the rest at y% loss. On the whole his loss is P%.  Then the quantity sold at x% profit is = ( y - P ) N Kg x + y
 and the quantity sold at y% loss is given by = ( x + P ) N Kg x + y

• Example :

• A trader has 50 kg of rice, a part of which he sells at 14% profit and the rest at 6% loss. On the whole his loss is 4%. What is the quantity sold at 14% profit and that at 6% loss?

• Detail Method :
Let the quantity sold at 14% profit be x kg. Then the quantity sold at 6% loss will be (50 - x) kg.
For a matter of convenience suppose that the price of rice is 1 rupee per kg.
Then price of x kg rice = Rs x and price of (50 - x) kg rice = Rs (50 - x)
Now, we have
14% profit of x + 6% loss of (50 - x)= 4% loss of 50
or, 14% of x - 6% of (50 - x) = 4% of 50
or, 14x 300 + 6x = 200
or, 20x = 100
∴ x = 5 kg
and 50 - x = 50 - 5 = 45 kg.
Therefore, the quantity sold at 14% profit = 5 kg and
the quantity sold at 6% loss = 45 kg.

• Ailigation Method :

 I Part II Part
(as there is loss on the whole)  14 (-)4 (-)6 2 18
∴ ratio of quantities sold at 14% profit and 6% loss
= 2 : 18 = 1 : 9
 ∴ quantity sold at 14% profit = 50 X 1 = 5 kg, 1 + 9
and sold at 6% loss = 50 - 5 = 45 kg.
Note: Numbers in the third line should always be +ve. That is why (-) 6 - (-)4 = -2 is not taken under consideration.

• Quicker Method : Here you can use direct formula :
 Quantity sold at 14% profit = ( 6 - 4 ) X 50 = 5 kg and 14 + 6
 the quantity sold at 6% loss = ( 14 + 4 ) X 50 = 45 kg. 14 + 6

Exercise :
1. A trader has 40 kg of rice, a part of which he sells at 28% profit and the rest at 12% loss. On the whole his loss is 8%. What is the quantity sold at 28% profit and that at 12% loss?
2. A trader has 48 kg of rice, a part of which he sells at 16% profit and the rest at 8% loss. On the whole his loss is 6%. What is the quantity sold at 16% profit and that at 8% loss?
3. A trader has 44 kg of rice, a part o f which he sells at 26% profit and the rest at 18% loss. On the whole his loss is 16%. What is the quantity sold at 26% profit and that at 18% loss?
4. Answers : 1 = 4 kg. 36 kg,   2 = 4 kg, 44 kg,   3 = 2 kg, 42 kg

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